Finite Group Databasebeta

← #14527740

Subgroups — #14527740

Order 2176.

256 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763435C211yesmaximal, characteristic
2#11762965C212yesmaximal, complemented
3#11762545C212yesmaximal, complemented
4#11762533C211yesmaximal, characteristic
5C17×#393C211yesmaximal, characteristic
6C22×Q8×C17C2211yescharacteristic
7C2×C22⋊C4×C17—21no—
8C2×C22⋊C4×C17C2211yescharacteristic
9C2×C22⋊Dic17—21no—
10C2×#63231—22no—
11#10587610—21no—
12#10587610C2212yes—
13#10587564—22no—
14#10587564—22no—
15C17×#95—21no—
16C17×#95C2211yescharacteristic
17#10587506—22no—
18C22⋊#3622C2212yes—
19C24×C17D411yescharacteristic
20C2×Q8×C17—22no—
21C2×Q8×C17—42no—
22C22×C4×C17—21no—
23C22×C4×C17—21no—
24C22×C4×C17D411yescharacteristic
25C22×C4×C17C2311yescharacteristic
26C22×Dic17—21no—
27C22×Dic17—21no—
28C2×Dic34—21no—
29C2×Dic34D412yes—
30C22⋊C4×C17—22no—
31C22⋊C4×C17—42no—
32C22⋊Dic17—22no—
33C22⋊Dic17—22no—
34#63231—44no—
35C4⋊Dic17—21no—
36C4⋊Dic17D412yes—
37#63225—22no—
38#63225—22no—
39C2×#3622—22no—
40C23×C17—21no—
41C23×C17—21no—
42C23×C17—41no—
43C23×C17—41no—
44C23×C17Q1612yescomplemented
45C23×C17C2×D411yescharacteristic
46Q8×C17—24no—
47Q8×C17—42no—
48C2×C4×C17—22no—
49C2×C4×C17—21no—
50C2×C4×C17—41no—
51C2×C4×C17—41no—
52C2×C4×C17—41no—
53C2×C4×C17—41no—
54C2×C4×C17—41no—
55C2×C4×C17—41no—
56C2×C4×C17—41no—
57C2×C4×C17C2×D412yes—
58C2×Dic17—22no—
59C2×Dic17—22no—
60C2×Dic17—81no—
61C2×Dic17—81no—
62Dic34—42no—
63Dic34—81no—
64#3622—42no—
65#1602—171nomaximal, Sylow
66C22×C17—21no—
67C22×C17—21no—
68C22×C17—22no—
69C22×C17—42no—
70C22×C17—41no—
71C22×C17—41no—
72C22×C17—81no—
73C22×C17#12011yescharacteristic
74C22×C17C2×Q1611yescharacteristic
75C22×C17#13711yescharacteristic
76C4×C17—22no—
77C4×C17—41no—
78C4×C17—42no—
79C4×C17—81no—
80Dic17—42no—
81Dic17—42no—
82#543—171no—
83#451—172no—
84#393D1711yescharacteristic
85#328—172no—
86#323—171no—
87C2×C17—21no—
88C2×C17—21no—
89C2×C17—42no—
90C2×C17#45111yescharacteristic
91C2×C17#45511yescharacteristic
92C2×C17B(4,𝔽2)11yescharacteristic
93C22×Q8C2×D1711yescharacteristic
94C2×Q16—342no—
95C22⋊Q8—172no—
96C22⋊Q8—342no—
97C22⋊Q8—342no—
98C22⋊Q8—341no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—341no—
101C2×C22⋊C4C2×D1711yescharacteristic
102#103—342no—
103C22⋊C8—172no—
104#95—21no—
105#95C2×D1711yescharacteristic
106C17#160211yescharacteristic, complemented, Sylow
107C24C22⋊D1711yescharacteristic
108C2×Q8—22no—
109C2×Q8—42no—
110C2×Q8—172no—
111C2×Q8—341no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—341no—
115C22×C4—341no—
116C22×C4C22⋊D1711yescharacteristic
117C22×C4C22×D1711yescharacteristic
118Q16—684no—
119C2×C8—342no—
120C4⋊C4—172no—
121C4⋊C4—342no—
122C4⋊C4—341no—
123C4⋊C4—342no—
124C22⋊C4—22no—
125C22⋊C4—42no—
126C22⋊C4—342no—
127C22⋊C4—342no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23#6323112yescomplemented
133C23C2×C22⋊D1711yescharacteristic
134Q8—24no—
135Q8—42no—
136Q8—682no—
137Q8—1361no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—342no—
148C2×C4—342no—
149C2×C4—1361no—
150C2×C4—1361no—
151C2×C4D4×D1712yes—
152C8—682no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#1058762311yescharacteristic
161C22C2×#6323111yescharacteristic
162C22#1058763711yescharacteristic
163C4—22no—
164C4—41no—
165C4—42no—
166C4—81no—
167C4—682no—
168C4—682no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#1176296411yescharacteristic
173C2#1176298611yescharacteristic
174C2#1176299511yescharacteristic
subgroupG/H
centerC22#10587623
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
FittingC17×#393C2
socleC22×C17#120
prime radicalC24×C17D4
hypercenter#393D17
hypocenterC17#1602
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #393

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #11762965C2semidirect
3. #11762545C2semidirect
44. C23×C17Q16semidirect
106. C17#1602semidirect
132. C23#63231semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#117634352
#117629652
#117625452
#117625332
C17×#3932
#160217